Non‐equivalent greedy and almost greedy bases in lp

Dilworth, S. J.; Hoffmann, Mark R.; Kutzarova, Denka

doi:10.1155/2006/368648

Cited by 9 publications

(7 citation statements)

References 14 publications

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“…The novelty in our approach has to be seen also in that our techniques are valid for the limit case p = 1. This nicely complements the main result from [8], where Dilworth et al showed that if 1 ≤ p < ∞ then ℓ p has a continuum of permutatively non-equivalent almost greedy bases.…”

Section: Introductionsupporting

confidence: 80%

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On certain subspaces of $${\ell _p}$$ for $${0<p\le 1}$$ and their applications to conditional quasi-greedy bases in p-Banach spaces

2020

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We construct for each 0 < p ≤ 1 an infinite collection of subspaces of ℓ p that extend the example of Lindenstrauss from [21] of a subspace of ℓ 1 with no unconditional basis. The structure of this new class of p-Banach spaces is analyzed and some applications to the general theory of L p -spaces for 0 < p < 1 are provided. The introduction of these spaces serves the purpose to develop the theory of conditional quasi-greedy bases in p-Banach spaces for p < 1. Among the topics we consider are the existence of infinitely many conditional quasi-greedy bases in the spaces ℓ p for p ≤ 1 and the careful examination of the conditionality constants of the "natural basis" of these spaces.

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Section: Introductionsupporting

confidence: 80%

“…The following was result was proved for the case ℓ 1 in [8] using completly different techniques. Corollary 6.11.…”

Section: For Every Concave Increasing Functionmentioning

confidence: 91%

On certain subspaces of $${\ell _p}$$ for $${0<p\le 1}$$ and their applications to conditional quasi-greedy bases in p-Banach spaces

2020

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“…For instance, in light of [24,Theorem 2.1], this technique can be applied to the L p -normalized Haar system to obtain a bidemocratic conditional quasi-greedy basis of L p ([0, 1]), p ∈ (1, 2) ∪ (2, ∞). Also, since, for the same values of p, the space ℓ p has a greedy basis which is non-equivalent to the canonical basis (see [14,Theorem 2.1]), this technique yields a bidemocratic conditional basis of ℓ p .…”

Section: Building Bidemocratic Conditional Quasi-greedy Basesmentioning

confidence: 99%

Bidemocratic bases and their connections with other greedy-type bases

Albiac¹,

Ansorena²,

Berasategui³

et al. 2021

Preprint

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In nonlinear greedy approximation theory, bidemocratic bases have traditionally played the role of dualizing democratic, greedy, quasi-greedy, or almost greedy bases. In this article we shift the viewpoint and study them for their own sake, just as we would with any other kind of greedy-type bases. In particular we show that bidemocratic bases need not be quasi-greedy, despite the fact that they retain a strong unconditionality flavor which brings them very close to being quasi-greedy. Our constructive approach gives that for each 1 < p < ∞ the space ℓ p has a bidemocratic basis which is not quasi-greedy. We also present a novel method for constructing conditional quasi-greedy bases which are bidemocratic, and provide a characterization of bidemocratic bases in terms of the new concepts of truncation quasi-greediness and partially democratic bases.

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“…In between these two extremes we come across spaces like ℓ p for 1 < p < ∞, p = 2, with a unique symmetric basis and a wealth of permutatively non-equivalent greedy bases (cf. [10]). The following question naturally arises: Problem 1.2.…”

Section: N=1mentioning

confidence: 99%

“…For instance, Smela proved that the L p -spaces for 1 < p < ∞, p = 2, and H 1 have infinitely many permutatively non-equivalent greedy bases [33] (cf. [10]).…”

Section: N=1mentioning

confidence: 99%

Banach spaces with a unique greedy basis

Albiac

Ansorena

Dilworth

et al. 2016

Journal of Approximation Theory

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The purpose of this article is to undertake an in-depth study of the properties of existence and uniqueness of greedy bases in Banach spaces. We show that greedy bases fail to exist for a range of neo-classical spaces within the family of Nakano and Orlicz sequence spaces and find the first-known cases of non-trivial spaces (i.e., different from c 0 , ℓ 1 , and ℓ 2) with a unique greedy basis. The variety and nature of those examples evince that a complete classification of Banach spaces with a unique greedy basis cannot be expected.

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Non‐equivalent greedy and almost greedy bases in l_p

Cited by 9 publications

References 14 publications

On certain subspaces of $${\ell _p}$$ for $${0<p\le 1}$$ and their applications to conditional quasi-greedy bases in p-Banach spaces

On certain subspaces of $${\ell _p}$$ for $${0<p\le 1}$$ and their applications to conditional quasi-greedy bases in p-Banach spaces

Bidemocratic bases and their connections with other greedy-type bases

Banach spaces with a unique greedy basis

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